
By Theorem 3.5 in [1]}, under Assumptions REF , REF  and REF , BSDE (REF ) admits a unique solution \((P_{1,N}^a(i),\Lambda ^a_{1,N}(i))_{i\in \mathcal {M}}\) , such that \((P^a_{1,N}(i), \ \Lambda ^a_{1,N}(i))\in L^\infty _{\mathcal {F}^W}(0, N; \mathbb {R})\times L^{2}_{\mathcal {F}^{W}}(0, N;\mathbb {R}^{n})\)  and \(P^a_{1,N}(i)\ge 0, \)  for all \(i\in \mathcal {M}\) . Same as before, we extend \((P^a_{1,N}(i),\Lambda ^a_{1,N}(i))\)  to \([0,+\infty )\)  by setting \(P^a_{1,N}(t,i)=0, \ \Lambda ^a_{1,N}(t,i)=0\)  for \(t>N\) .
